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Laplace Transforms SECTION 5: LAPLACE TRANSFORMS ESE 330 – Modeling & Analysis of Dynamic Systems 2 Introduction – Transforms This section of notes contains an introduction to Laplace transforms. This should mostly be a review of material covered in your differential equations course. K. Webb ESE 330 Transforms 3 What is a transform? A mapping of a mathematical function from one domain to another A change in perspective not a change of the function Why use transforms? Some mathematical problems are difficult to solve in their natural domain Transform to and solve in a new domain, where the problem is simplified Transform back to the original domain Trade off the extra effort of transforming/inverse- transforming for simplification of the solution procedure K. Webb ESE 330 Transform Example – Slide Rules 4 Slide rules make use of a logarithmic transform Multiplication/division of large numbers is difficult Transform the numbers to the logarithmic domain Add/subtract (easy) in the log domain to multiply/divide (difficult) in the linear domain Apply the inverse transform to get back to the original domain Extra effort is required, but the problem is simplified K. Webb ESE 330 5 Laplace Transforms K. Webb ESE 330 Laplace Transforms 6 An integral transform mapping functions from the time domain to the Laplace domain or s-domain ℒ Time-domain functions are↔ functions of time, Laplace-domain functions are functions of is a complex variable = + K. Webb ESE 330 Laplace Transforms – Motivation 7 We’ll use Laplace transforms to solve differential equations Differential equations in the time domain difficult to solve Apply the Laplace transform Transform to the s-domain Differential equations become algebraic equations easy to solve Transform the s-domain solution back to the time domain Transforming back and forth requires extra effort, but the solution is greatly simplified K. Webb ESE 330 Laplace Transform 8 Laplace Transform: = = (1) ∞ − Unilateralℒ or one -sided transform∫0 Lower limit of integration is = 0 Assumed that the time domain function is zero for all negative time, i.e. = 0, < 0 K. Webb ESE 330 9 Laplace Transform Properties In the following section of notes, we’ll derive a few important properties of the Laplace transform. K. Webb ESE 330 Laplace Transform – Linearity 10 Say we have two time-domain functions: and Applying the 1transform 2definition, (1) + = + ∞ − ℒ 1 2 � 1 2 = 0 + ∞ ∞ − − � 1 � 2 = 0 + 0 ∞ ∞ − − 1 2 = �0 + �0 � ℒ 1 � ℒ 2 + = + (2) The ℒLaplace1 transform 2is a linear operation1 2 K. Webb ESE 330 Laplace Transform of a Derivative 11 Of particular interest, given that we want to use Laplace transform to solve differential equations = ∞ − Use integrationℒ bẏ parts�to0 evaluatė = Let = ∫ and − ∫ = then − ̇ = and = − K. Webb − ESE 330 Laplace Transform of a Derivative 12 = ∞ ∞ − − ℒ ̇ �0 − �0 − = 0 0 + = 0 + ∞ − − �0 − ℒ The Laplace transform of the derivative of a function is the Laplace transform of that function multiplied by minus the initial value of that function = (0) (3) ℒ ̇ − K. Webb ESE 330 Higher-Order Derivatives 13 The Laplace transform of a second derivative is = 0 0 (4) 2 ℒ ̈ − − ̇ In general, the Laplace transform of the derivative of a function is given by = 0 0 0 (5) −1 −2 −1 ℒ − − ̇ − ⋯ − K. Webb ESE 330 Laplace Transform of an Integral 14 The Laplace Transform of a definite integral of a function is given by = (6) 1 ℒ ∫0 Differentiation in the time domain corresponds to multiplication by in the Laplace domain Integration in the time domain corresponds to division by in the Laplace domain K. Webb ESE 330 Laplace Transforms of Common 15 Functions Next, we’ll derive the Laplace transform of some common mathematical functions K. Webb ESE 330 Unit Step Function 16 A useful and common way of characterizing a linear system is with its step response The system’s response (output) to a unit step input The unit step function or Heaviside step function: 0, < 0 1 = 1, 0 � ≥ K. Webb ESE 330 Unit Step Function – Laplace Transform 17 Using the definition of the Laplace transform 1 = 1 = ∞ ∞ − − ℒ � � 1 0 1 0 1 = = 0 = ∞ − − �0 − − The Laplace transform of the unit step 1 = (7) 1 Note that the unilateralℒ Laplace transform assumes that the signal being transformed is zero for < 0 Equivalent to multiplying any signal by a unit step K. Webb ESE 330 Unit Ramp Function 18 The unit ramp function is a useful input signal for evaluating how well a system tracks a constantly- increasing input The unit ramp function: 0, < 0 = , 0 � ≥ K. Webb ESE 330 Unit Ramp Function – Laplace Transform 19 Could easily evaluate the transform integral Requires integration by parts Alternatively, recognize the relationship between the unit ramp and the unit step Unit ramp is the integral of the unit step Apply the integration property, (6) 1 1 = 1 = ℒ ℒ � � 0 = (8) 1 2 K. Webb ℒ ESE 330 Exponential – Laplace Transform 20 = − Exponentials are common components of the responses of dynamic systems = = ( ) ∞ ∞ − − − − + ℒ � � 0 01 = = 0 − ++ ∞ + − �0 − − = (9) − 1 K. Webb ℒ + ESE 330 Sinusoidal functions 21 Another class of commonly occurring signals, when dealing with dynamic systems, is sinusoidal signals – both sin and cos = sin Recall Euler’s formula = cos + sin From which it follows that sin = − − K. Webb ESE 330 2 Sinusoidal functions 22 1 sin = ∞ − − ℒ 1 �0 − = 2 ∞ − − − + 1 �0 − 1 = 2 ∞ ∞ − − − + � − � 21 0 21 0 = − − ∞ − ++ ∞ 1 1 �0 −1 1 �01 = 2 0−+ − 20 +− = + + 2 − 2 2 2 − 2 2 sin = (10) 2 2 K. Webb ℒ + ESE 330 Sinusoidal functions 23 It can similarly be shown that cos = (11) 2 2 ℒ + Note that for neither sin( ) nor cos is the function equal to zero for < 0 as the Laplace transform assumes Really, what we’ve derived is 1 sin and 1 cos K. Webb ℒ � ℒ � ESE 330 24 More Properties and Theorems K. Webb ESE 330 Multiplication by an Exponential, 25 − We’ve seen that = − 1 What if another functionℒ is +multiplied by the decaying exponential term? = = ∞ ∞ − − − − + ℒ � � This is just the Laplace0 transform of0 with replaced by + = + (12) − K. Webb ℒ ESE 330 Decaying Sinusoids 26 The Laplace transform of a sinusoid is sin = + ℒ 2 2 And, multiplication by an decaying exponential, , results in a substitution of + for , so − sin = + + − and ℒ 2 2 + cos = + + − 2 2 K. Webb ℒ ESE 330 Time Shifting 27 Consider a time-domain function, To Laplace transform we’ve assumed = 0 for < 0, or equivalently multiplied by 1() To shift by an amount, , in time, we must also multiply by a shifted step function, 1 K. Webb − ESE 330 Time Shifting – Laplace Transform 28 The transform of the shifted function is given by 1 = ∞ − ℒ − � − � − Performing a change of variables, let = and = The transform becomes − 1 = = = ∞ ∞ ∞ − + − − − − ℒ � � � � A shift by in the0 time domain corresponds0 to multiplication0 by in the Laplace domain − 1 = (13) − ℒ − � − K. Webb ESE 330 Multiplication by time, 29 The Laplace transform of a function multiplied by time: = ( ) (14) Consider a unitℒ ramp⋅ function:− 1 1 = 1 = = ℒ ℒ ⋅ − 2 Or a parabola: = = = 2 1 2 2 3 In general ℒ ℒ ⋅ − ! = ℒ +1 K. Webb ESE 330 Initial and Final Value Theorems 30 Initial Value Theorem Can determine the initial value of a time-domain signal or function from its Laplace transform 0 = lim (15) →∞ Final Value Theorem Can determine the steady-state value of a time-domain signal or function from its Laplace transform = lim (16) ∞ →0 K. Webb ESE 330 Convolution 31 Convolution of two functions or signals is given by = Result is a function of∗ time �0 − is flipped in time and shifted by Multiply the flipped/shifted signal and the other signal Integrate the result from = 0 … May seem like an odd, arbitrary function now, but we’ll later see why it is very important Convolution in the time domain corresponds to multiplication in the Laplace domain = (17) K. Webb ℒ ∗ ESE 330 Impulse Function 32 Another common way to describe a dynamic system is with its impulse response System output in response to an impulse function input Impulse function defined by = 0, 0 = 1 ≠ ∞ � An infinitely−∞ tall, infinitely narrow pulse K. Webb ESE 330 Impulse Function – Laplace Transform 33 To derive , consider the following function 1 , 0 ℒ= 0, < 0 or≤ ≤> 0 �0 0 Can think of as the sum of two step functions: 1 1 = 1 1 − − 0 The transform of the first term0 is 0 1 1 1 = ℒ Using the time-shifting property,0 the second0 term transforms to 1 1 = −0 0 ℒ − 0 − − 0 K. Webb ESE 330 Impulse Function – Laplace Transform 34 In the limit, as 0, , so 0 → →= lim ℒ 0→01ℒ = lim −0 − ℒ 0→0 Apply l’Hôpital’s rule 0 1 = lim −0 = lim = −0 − 0 ℒ 0→0 0→0 0 The Laplace transform of an impulse0 function is one = 1 (18) K. Webb ℒ ESE 330 Common Laplace Transforms 35 1 sin + + − 1 +2 2 1 cos + + − 1 2 2 ( ) (0) 2! ̇ − 0 0 2 1+1 ̈ −1 − ̇ + − 1 � 0 + + − − 2 sin 1 + − 2 2 cos − � − + 2 2 K. Webb � − ESE 330 Example – Piecewise Function Laplace Transform 36 Determine the Laplace transform of a piecewise function: A summation of functions with known transforms: Ramp Pulse – sum of positive and negative steps Transform is the sum of the individual, known transforms K. Webb ESE 330 Example – Piecewise Function Laplace Transform 37 Treat the piecewise function as a sum of individual functions = + 1 2 Time-shifted, gated ramp 1 Time-shifted pulse 2 Sum of staggered positive and negative steps K.
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